On theh- andp-Versions of the Extrapolated Gordon’s Projector with Applications to Elliptic Equations

Abstract

This paper outlines a new application of Cordon’s blending technique for the finite element approximation of elliptic boundary value problems. The algebraic structure of the discrete blended interpolation projector with its first- (or coarse-) and second- (or fine-) discretization levels suggests making linear combinations of independent calculations. For the interpolation of smooth data, each of the separate calculations yields a low-order result while their combination gives a high-order result. It is conjectured that this property holds for the approximation of regular boundary value problems (BVPs). The algorithm might therefore be viewed as an extrapolation procedure. Two different versions of the algorithm are proposed, which relate to the so-called h- and p-versions of finite element approximations. The computational complexities compare favorably with classical schemes. The implementation on parallel computers is discussed. Numerical results for some bivariate problems (both regular and singular) are presented. They indicate that for smooth problems, the algorithms behave as expected.